Hex collet block.
But really you can just buy some hex rod, and shave it down as needed to tightly fit the socket cap screw. The more important bit is the concentricity you'll get turning part of the hex rod down to a round shaft in the lathe.
Ah! I think we've got a winner here! I don't have a collet block (yet. Budgets are inconvenient, aren't they!), and my mill is probably less precise than the screw I've already got anyway! It's a well-loved Rong-Fu round column benchtop machine. Will probably upgrade sometime in the distant future, but a surface grinder is on my list before a Bridgeport. And a concrete shop is on my list before that! Anyway, back to the idea at hand!
I'm puzzled by what you mean by "The more important bit is the concentricity you'll get turning part of the hex rod down to a round shaft in the lathe."
Are you saying that I buy 3/16 hex stock (the size I need for driving my screw) and turn part of it down, put the round part in my chuck while noting which side is "up" so the runout cancels itself out?
I'm also contemplating getting 3/16 hex stock and machining a steep taper on the very end of it so it contacts the head of the screw in the shallow taper inside the socket while also providing enough contact with the socket to drive the screw.
Just to be clear and restating my goals...
I'm center drilling the threaded end of the screw while holding the threads in the chuck. I move the screw out of the chuck to be driven by a hex tool bit (either something I make, or something made across the sea) and held by a live center and finishing the machining there. I'm taking material off the diameter of the head as well as the shank, and my goal is to have everything be concentric with the socket in the head and the center drilled shank.
Did I miss anything? I did mention that I am a beginner, right? In case you couldn't tell.
Also, I did try the Torx bit. It is just a hair tighter. It is about .002 bigger across the flats than the hex bit. If all else fails, I might go for that, but it is still too wobbly for my tastes.